What is the harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$?

If $\alpha, \beta$ are the roots of the equation $lx^2 + mx + n = 0$,then the equation whose roots are $\alpha^3\beta$ and $\alpha\beta^3$ is

Let $\alpha$ and $\beta$ be two distinct roots of the equation $x^3 + 3x^2 - 1 = 0$. The equation which has $(\alpha \beta)$ as its root is equal to

If the roots of the equation $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$,then the value of $\alpha\beta^2 + \alpha^2\beta + \alpha\beta$ will be

Suppose the quadratic polynomial $p(x)=ax^2+bx+c$ has positive coefficients $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$,then what could be the possible value of $\alpha+\beta+\alpha\beta$ if $0 \leq \alpha+\beta+\alpha\beta \leq 8$?

If $\alpha, \beta, \gamma$ are the roots of $x^3 - x - 2 = 0$,then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-